MOnaliZumab in Combination With durvAlumab (MEDI4736) for tRreatmenT of Small Cell Lung Cancer

General Inclusion Criteria: 1. Written informed consent and HIPAA authorization for release of personal health information prior to registration. Note: HIPAA authorization may be included in the informed consent or obtained separately. 2. Age ≥ 18 years at the time of consent. 3. Demonstrate adequate organ function. All screening labs to be obtained within 28 days prior to registration. * Absolute Neutrophil Count (ANC) \> 1500mm\^3 * Hemoglobin ≥ 9 g/dL * Platelet Count (PLT) ≥ 100,000 per mm3 * Calculated creatinine clearance ≥ 40 mL/min * Bilirubin ≤ 1.5 × upper limit of normal (ULN); subjects with confirmed Gilbert's syndrome (persistent or recurrent hyperbilirubinemia that is predominantly unconjugated in the absence of hemolysis or hepatic pathology), may be allowed with sponsor-investigator approval. * Apsartate aminotransferase (AST) ≤ 2.5 x institutional upper limit of normal unless liver metastases are present, in which case it must be ≤5x ULN * Alanine aminotransferase (ALT) ≤ 2.5 x institutional upper limit of normal unless liver metastases are present, in which case it must be ≤5x ULN 4. Females of childbearing potential must have a negative serum pregnancy test at screening. 5. Females of childbearing potential and male subjects must be willing to abstain from heterosexual intercourse or to use an effective method(s) of contraception. 6. Life expectancy of ≥ 12 weeks. 7. Patients with evidence of chronic hepatitis B virus (HBV) infection, the HBV viral load must be undetectable on suppressive therapy, if indicated. Patients with a history of hepatitis C virus (HCV) infection must have been treated and cured. For patients with HCV infection who are currently on treatment, the HCV viral load must be undetectable through PCR to be eligible for this trial. Testing is not required for screening unless mandated by local authorities. Local guidelines for testing should be followed. Extensive Stage Specific Inclusion Criteria: 1. Histologically or cytologically confirmed diagnosis of small cell lung cancer: \- Extensive disease (American Joint Committee on Cancer Stage (8th edition) IV SCLC \[T any, N any, M1 a/b\]), OR T3-4 disease due to multiple lung nodules that are too extensive or have tumor/nodal volume that is too large to be encompassed in a tolerable radiation plan. 2. No prior systemic therapy for small-cell lung cancer, with the following exceptions: Up to one cycle of platinum doublet chemotherapy with or without durvalumab is allowed up to 4 weeks prior to registration on this study. Patients with irreversible toxicity not reasonably expected to be exacerbated by treatment with durvalumab and monalizumab may be included only after consultation with the sponsor-investigator. Patients should not have received Trilaciclib. 3. Measurable disease according to RECIST v1.1. 4. Subjects with treated brain metastasis or untreated asymptomatic brain metastasis that is clinically stable per investigator discretion and not requiring systemic steroids for ≥ 7 days. NOTE: Prophylactic cranial radiation (PCI) is allowed per investigator's discretion. 5. ECOG Performance Status of 0-2. Limited Stage Specific Inclusion Criteria: 1. Histologically or cytologically confirmed diagnosis of small cell lung cancer: \- Limited-stage disease (American Joint Committee on Cancer Stage (8th edition) I-III SCLC \[T any, N any, M0\]) 2. Has received platinum (cis- or carboplatin) and etoposide chemotherapy (4 cycles preferred; 3 cycles allowed if disease control is achieved and no additional benefit is expected with an additional cycle of chemotherapy in the opinion of the investigator) administered concurrently with radiation (60-66Gy daily or 45Gy BID). Radiation should have started no later than end of cycle 2 of chemotherapy. 3. Non-progressive disease following completion of chemo-radiation. 4. No evidence of brain metastasis. NOTE: PCI is allowed per investigator's discretion. 5. Ability to start study treatment within 56 days of completing chemo-radiation, counting from whichever ends later 6. ECOG Performance Status of 0-1. Exclusion Criteria: 1. Body weight ≤ 40 kg. 2. Active infection requiring intravenous antibiotic therapy. 3. Known allergy or hypersensitivity to any of the study drugs or any of the study drug excipients. 4. Major surgical procedure (as defined by the investigator) within 28 days prior to the first dose of study treatment. NOTE: Local surgery of isolated lesions for palliative intent is acceptable. 5. History of active primary immunodeficiency. 6. Known to have tested positive for human immunodeficiency virus (HIV) (positive HIV 1/2 antibodies) or active tuberculosis infection (clinical evaluation that may include clinical history, physical examination and radiographic findings, or tuberculosis testing in line with local practice). 7. Presence of neurologic paraneoplastic syndrome. 8. Active or prior documented autoimmune or inflammatory disorders (including inflammatory bowel disease \[e.g., ulcerative colitis or Crohn's disease\], systemic lupus erythematosus, sarcoidosis, Wegener syndrome \[granulomatosis with polyangiitis\], rheumatoid arthritis, hypophysitis, uveitis, etc). The following are exceptions to this criterion: * Patients with vitiligo or alopecia * Patients with hypothyroidism (e.g., following Hashimoto syndrome) stable on hormone replacement * Any chronic skin condition that does not require systemic therapy * Patients without active disease in the last 2 years may be included but only after consultation with the study physician * Patients with celiac disease controlled by diet alone 9. Receipt of live attenuated vaccine within 30 days prior to the first dose of study treatment. NOTE: Subjects, if enrolled, should not receive live vaccine whilst receiving study treatment and up to 30 days after the last dose of study treatment. 10. Uncontrolled intercurrent illness, including but not limited to, ongoing or active infection, symptomatic congestive heart failure, uncontrolled hypertension, unstable angina pectoris, cardiac arrhythmia, interstitial lung disease, serious chronic gastrointestinal conditions associated with diarrhea, or psychiatric illness/social situations that would limit compliance with study requirement, substantially increase risk of incurring AEs or compromise the ability of the patient to give written informed consent. 11. Pregnant or breastfeeding (NOTE: breast milk cannot be stored for future use while the mother is being treated on study). 12. Patients with a prior or concurrent malignancy whose natural history or treatment has the potential to interfere with the safety or efficacy assessment of the investigational regimen, per investigator discretion. 13. History of leptomeningeal carcinomatosis. 14. History of allogeneic organ transplantation. 15. Treatment with any investigational drug within 28 days prior to registration or concurrent enrolment in another clinical study, unless observational in nature. 16. Current or prior use of immunosuppressive medication within 7 days before the first dose of monalizumab and durvalumab (applicable to 'on study' durvalumab for MOZART-ES cohort who may have received prior one dose of durvalumab). The following are exceptions to this criterion: * Intranasal, inhaled, topical steroids, or local steroid injections (e.g., intra articular injection) * Systemic corticosteroids at physiologic doses not to exceed 10 mg/day of prednisone or its equivalent * Steroids as premedication for hypersensitivity reactions (e.g., CT scan premedication), and for prevention of chemotherapy induced nausea/vomiting per institutional standards. 17. Specific for MOZART-ES cohort: Patients who have received prior one dose of durvalumab along with chemotherapy: * Must not have experienced a toxicity that led to permanent discontinuation of prior immunotherapy. * Must not have experienced a ≥Grade 3 immune related AE or an immune related neurologic or ocular AE of any grade while receiving prior immunotherapy. NOTE: Patients with endocrine AE of ≤ Grade 2 are permitted to enroll if they are stably maintained on appropriate replacement therapy and are asymptomatic. * Must not have required the use of additional immunosuppression other than corticosteroids for the management of an AE and not currently require maintenance doses of \> 10 mg prednisone or equivalent per day.